As the Sun fuses hydrogen, the core begins to become "clogged up" with helium "ash", which would tend to damp out the fusion reaction. The helium ash also increases the density of the core. The result is that the temp and pressure goes up which increases the fusion rate which tends to overcompensate for the helium build up. The increased fusion rate causes the star's luminosity and radius to go up.

As the Sun fuses hydrogen, the core begins to become "clogged up" with helium "ash", which would tend to damp out the fusion reaction. The helium ash also increases the density of the core. The result is that the temp and pressure goes up which increases the fusion rate which tends to overcompensate for the helium build up. The increased fusion rate causes the star's luminosity and radius to go up.

I know that it sounds odd, but consider this: helium is "heavier" than hydrogen and would tend to "sink" to the center of the core. This tends to snuff out the fusion at the center, but at the same time increases the density of the core. The core condenses, and as it does so, the temp and pressure goes up so that fusion can take place in the hydrogen shell surrounding the helium clogged core. So while fusion at the center goes down, it is more than made up for by an increase in fusion in the region surrounding the center.

Additionally, you are adding electrons to the star, which increases the opacity. That also "clogs up the works", in this case the photon transport, which also increases the core temperature and increases the fusion rate. So as the opacity increases, the star gets brighter.

It states: "...the composition of the core changes slowly as nucleosynthesis reduces the abundance of hydrogen. Decreases in pressure differences allow gravity to increase the density, temperature, and radius of the thermonuclear core."

I guess I'm just slow, but what implies decreases in pressure differences, and how dos this lead to an increase in density, temp, and radius of the reactive core?

Don’t be so hard on yourself, Phrak. It took the combined genius of the world millennia to figure this out. A star is sort of a wrestling match between the fusion at the core causing an outward pressure that almost rips it apart and the gravity of its mass trying to crush it into a dense point. The hydrogen at the core is converted into helium by fusion (a process known as nucleosynthesis). The core is not yet hot enough to convert the helium into say, carbon, so the helium collects at the core. This slows the fusion of hydrogen and causes the outward pressure of the fusion explosion to lessen. Relatively speaking, the core cools. With less outward pressure, gravity starts to win and compresses the core more than it had previously. This increases the density of the core as gravity tries to crush everything together. In turn, increased density causes the temperature to rise and the volume that is undergoing nucleosynthesis increases as well, i.e., a greater radius of the core fuses.

"But why does the star pulsate at all??
The textbook has a rather involved description involving the properties of a layer of ionized helium in the star's envelope. A simpler, stripped-down explanation goes something like this:
When a Cepheid is compressed, it becomes opaque.
Photons are trapped inside, heating the gas and increasing its pressure.
The high-pressure gas expands, becoming transparent.
Photons escape, the gas cools, the pressure drops.
As the pressure drops, the Cepheid is compressed by gravity.
This cycle repeats as long as the Cepheid (or the RR Lyrae, which pulsates by the same mechanism) is in the instability strip, the region of the H-R diagram where stars are unstable to pulsation. "